Optimal pricing for a heterogeneous portfolio for a given risk factor and convex distance measure

Consider a portfolio containing heterogeneous risks. The premiums of the policyholders might not cover the amount of the payments which an insurance company pays the policyholders. When setting the premium, this risk has to be taken into consideration. On the other hand the premium that the insured pays has to be fair. This fairness is measured by a function of the difference between the risk and the premium paid—we call this function a distance function. For a given small probability of insolvency, we find the premium for each class, such that the distance function is minimized. Next we formulate and solve the dual problem, which is minimizing the insolvency probability under the constraint that the distance function does not exceed a given level. This paper generalizes a previous paper [Zaks, Y., Frostig, E., Levikson, B., 2006. Optimal pricing of a heterogeneous portfolio for a given risk level. Astin Bull. 36 (1), 161–185] where only a square distance function was considered.     

A new approximation operator of the Arato-Renyi type

In this paper a new approximation operator is introduced and its properties are studied. Special cases of this operator are the well-known Szàsz power-series approximation operator and its generalization by D. Leviatan. The behaviour of the new approximation operator at points of continuity and discontinuity is investigated by using probabilistic tools as the Chebishev inequality and Liapounov’s central limit theorem. Such probabilistic methods of proof simplify the proofs and give better understanding of the approximation mechanism.     

Supermodular Comparison of Time-to-Ruin Random Vectors

This paper studies time-to-ruin random vectors for multivariate risk processes. Two cases are considered: risk processes with independent increments and risk processes evolving in a common random environment (e.g., because they share the same economic conditions). As expected, increasing the dependence between the risk processes increases the dependence between their respective time-to-ruin random variables. This article is dedicated to the memory of our beloved friend Benjamin Zeev Levikson who passed away on July 16, 2005.     

An Algorithm Evaluating Generalized Life Insurance Programs

We study generalized life insurance (GLI) models in continuous time. These models are presented as non-homogeneous Semi-Markov processes and studied directly as such. We give an algorithm, based on recursive integral scheme, finding the expected present value of premium payments and of benefits outgo, thus enabling us to find the annual premium. An algorithm based on this method is applied numerically using real data set to calculate the above quantities for a GLI contract.